# Stiffness of Metal

The resistance of a metal to elastic deformation is called its stiffness of metal. Since the deflections produced in many products by operating loads may affect the workability of the components, the stiffness of the material is often a limiting design factor.

Peculiarly enough, the stiffness of a metal is varied very little by alloying with other elements. If a steel beam sags too much under load, nothing will be gained by changing to a stronger alloy steel. The strength will be increased, but if the load is the same, the beam will still sag about the same amount. The remedy is to increase the cross-section of the beam.

The stiffness of a metal is measured by the moduli of elasticity in tension, compression, and shear, by Poisson’s ratio, and by the bulk modulus of elasticity.

## Modulus of Elasticity in Tension and Compression

The modulus of elasticity is a measure of the elastic deformation of a metal when stressed in tension or compression within the proportional limit. Its value is equal to the ratio of stress to strain and corresponds to the initial slope of the stress-strain curve.

When strain is small, the ratio of the longitudinal stress to the corresponding longitudinal strain is called the Young’s modulus of the material of the body.

When strain is small, the ratio of the normal stress to the volume strain is called the bulk modulus of the material of the body.

## Poisson’s Ratio

When two equal and opposite forces are applied to a body along a certain direction, the body elongates along that direction. At the same time, it also contracts along the perpendicular directions.

The fractional change in the direction along which the forces have been applied is called the longitudinal strain, while the fractional change in a perpendicular direction is called the lateral strain.

The ratio of the lateral strain to the longitudinal strain is known as the Poisson’s ratio. It is a constant for the material of the body.

As shown in Figure #1, a wire of original length ‘L’ and diameter ‘D’ is subjected to equal and opposite forces ‘F’, ‘F’, along its length. If the length increases by ‘∆L’ and the diameter decreases by delta ‘∆D’, then

Longitudinal strain = ∆L/L
and lateral strain = ∆D/D

The Poisson’s ratio ‘σ’ of the material of the wire is
σ = lateral strain/longitudinal strain = (∆D/D)/(∆L/L)

‘σ’ being a ratio of two types of strains, has no unit and is dimensionless. Theoretically, the value of ‘σ’ should be less than 0.5. For most solids, it lies between 0.25 and 0.35.

## Shear and Modulus of Rigidity

When a body is acted upon by an external force tangential to a surface of the body, the opposite surface being kept fixed, it suffers a change in shape; its volume remaining unchanged. Then the body is said to be ‘sheared’.

The ratio of the displacement of a layer in the direction of the tangential force and the distance of that layer from the fixed surface is known as ‘shear’ or ‘shearing strain’ and the tangential force acting per unit area of the surface is known as the ‘shearing stress’.

For small strain, the ratio of the shearing stress to the shearing strain is known as the ‘modulus of rigidity’ of the material of the body. It is denoted by ‘ἠ’

Let us consider a cube shown in Figure #3 whose lower surface is fixed. When a tangential force F is applied at its upper surface CDHG, then all the layers parallel to CDHG are displaced in the direction of the force. The displacement of a layer is proportional to its distance from the fixed surface. The cube thus takes the new form ABD’C’EFH’G’ as shown in Figure #4.

The angle through which the line AC or BD, initially perpendicular to the fixed surface, is turned is known as the ‘angle of shear’ or ‘shearing strain’.

If θ is small, then

θ = Tan θ
= DD’/BD
= Displacement of the upper surface/Distance of the upper surface from the fixed surface

Clearly, a shear is numerically equal to the ratio of the displacement of any layer to the distance of the layer from the fixed surface.

If ‘A’ be the area of the upper surface (CDHG), then
shearing stress = F/A

Modulus of rigidity of the material of the cube is
ἠ = Shearing stress/ Shear = (F/A)/θ = F/Aθ